View source for Abelian group

{{short description|Commutative group (mathematics)}}

In [[Mathematics|mathematics]], an '''abelian group''', also called a '''commutative group''', is a [[Group (mathematics)|group]] in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the [[Integer|integer]]s and the [[Real number|real number]]s form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician [[Biography:Niels Henrik Abel|Niels Henrik Abel]].<ref>{{Harvard citation text|Jacobson|2009}} p. 41</ref>

The concept of an abelian group underlies many fundamental [[Algebraic structure|algebraic structure]]s, such as [[Field (mathematics)|fields]], [[Ring (mathematics)|rings]], [[Vector space|vector space]]s, and [[Algebra over a field|algebras]]. The theory of abelian groups is generally simpler than that of their [[Non-abelian group|non-abelian]] counterparts, and finite abelian groups are very well understood and fully classified.

{{Group theory sidebar|Basics}}

== Definition ==
An abelian group is a [[Set (mathematics)|set]] <math>A</math>, together with an [[Binary operation|operation]] <math>\cdot</math> that combines any two [[Element (mathematics)|elements]] <math>a</math> and <math>b</math> of <math>A</math> to form another element of <math>A,</math> denoted <math>a \cdot b</math>. The symbol <math>\cdot</math> is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, <math>(A, \cdot)</math>, must satisfy four requirements known as the ''abelian group axioms'' (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation is ''defined'' for any ordered pair of elements of {{mvar|A}}, that the result is ''[[Well-defined expression|well-defined]]'', and that the result ''[[Element (mathematics)#Notation and terminology|belongs to]]'' {{mvar|A}}):

;Associativity: For all <math>a</math>, <math>b</math>, and <math>c</math> in <math>A</math>, the equation <math>(a \cdot b)\cdot c = a \cdot (b \cdot c)</math> holds.
;Identity element: There exists an element <math>e</math> in <math>A</math>, such that for all elements <math>a</math> in <math>A</math>, the equation <math>e \cdot a = a \cdot e = a</math> holds.
;Inverse element: For each <math>a</math> in <math>A</math> there exists an element <math>b</math> in <math>A</math> such that <math>a \cdot b = b \cdot a = e</math>, where <math>e</math> is the identity element.
;Commutativity: For all <math>a</math>, <math>b</math> in <math>A</math>, <math>a \cdot b = b \cdot a</math>.

A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".<ref>Ramík, J., ''Pairwise Comparisons Method: Theory and Applications in Decision Making'' (Cham: [[Company:Springer Nature|Springer Nature Switzerland]], 2020), [https://books.google.com/books?id=2qDSDwAAQBAJ&pg=PA11 p. 11].</ref>{{rp|11}}

== Facts ==
=== Notation ===
There are two main notational conventions for abelian groups – additive and multiplicative.

{| class="wikitable" style="margin: 1em auto 1em auto"
|-
! Convention
! Operation
! Identity
! Powers
! Inverse
|-
! Addition
| <math>x + y</math> || 0 || <math>nx</math> || <math>-x</math>
|-
! Multiplication
| <math>x \cdot y</math> or <math>xy</math> || 1
| <math>x^n</math>
| <math>x^{-1}</math>
|}

Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for [[Module (mathematics)|module]]s and [[Ring (mathematics)|ring]]s. The additive notation may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered, some notable exceptions being [[Near-ring|near-ring]]s and [[Partially ordered group|partially ordered group]]s, where an operation is written additively even when non-abelian.<ref>[[Biography:Maurice Auslander|Auslander, M.]], & [[Biography:David Buchsbaum|Buchsbaum, D.]], ''Groups, Rings, Modules'' (Mineola, NY: [[Social:Dover Publications|Dover Publications]], 1974), [https://books.google.com/books?id=MVEuBAAAQBAJ&pg=PA28 pp. 28–29].</ref>{{rp|28–29}}<ref>Stanojkovski, M., ''Intense Automorphisms of Finite Groups'' (Providence, RI: American Mathematical Society, 2021) [https://books.google.com/books?id=d_NUEAAAQBAJ&newbks=1&newbks_redir=0&printsec=frontcover&pg=PA9 pp. 9–14].</ref>{{rp|9–14}}

=== Multiplication table ===
To verify that a [[Finite group|finite group]] is abelian, a table (matrix) – known as a [[Cayley table]] – can be constructed in a similar fashion to a [[Multiplication table|multiplication table]].<ref>Isaev, A. P., & [[Biography:Valery Rubakov|Rubakov, V. A.]], ''Theory of Groups and Symmetries: Finite Groups, Lie Groups, and Lie Algebras'' ([[Place:Singapore|Singapore]]: [[Company:World Scientific|World Scientific]], 2018), [https://books.google.com/books?id=7sFUDwAAQBAJ&pg=PA10 p. 10].</ref>{{rp|10}} If the group is <math>G = \{g_1 = e, g_2, \dots, g_n \}</math> under the {{nowrap|operation <math>\cdot</math>,}} the {{nowrap|<math>(i, j)</math>-th}} entry of this table contains the product <math>g_i \cdot g_j</math>.

The group is abelian [[If and only if|if and only if]] this table is symmetric about the main diagonal. This is true since the group is abelian [[If and only if|iff]] <math>g_i \cdot g_j = g_j \cdot g_i</math> for all <math>i, j = 1, ..., n</math>, which is iff the <math>(i, j)</math> entry of the table equals the <math>(j, i)</math> entry for all <math>i, j = 1, ..., n</math>, i.e. the table is symmetric about the main diagonal.

== Examples ==
* For the [[Integer|integer]]s and the operation [[Addition|addition]] <math>+</math>, denoted <math>(\mathbb{Z}, +)</math>, the operation + combines any two integers to form a third integer, addition is associative, zero is the [[Additive identity|additive identity]], every integer <math>n</math> has an [[Additive inverse|additive inverse]], <math>-n</math>, and the addition operation is commutative since <math>n + m = m + n</math> for any two integers <math>m</math> and <math>n</math>.
* Every [[Cyclic group|cyclic group]] <math>G</math> is abelian, because if <math>x</math>, <math>y</math> are in <math>G</math>, then <math>xy = a^ma^n = a^{m+n} = a^na^m = yx</math>. Thus the [[Integer|integer]]s, <math>\mathbb{Z}</math>, form an abelian group under addition, as do the [[Modular arithmetic|integers modulo <math>n</math>]], <math>\mathbb{Z}/n \mathbb{Z}</math>.
* Every [[Ring (mathematics)|ring]] is an abelian group with respect to its addition operation. In a [[Commutative ring|commutative ring]] the invertible elements, or [[Unit (ring theory)|units]], form an abelian [[Multiplicative group|multiplicative group]]. In particular, the [[Real number|real number]]s are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.
* Every [[Subgroup|subgroup]] of an abelian group is [[Normal subgroup|normal]], so each subgroup gives rise to a [[Quotient group|quotient group]]. Subgroups, quotients, and [[Direct sum of groups|direct sums]] of abelian groups are again abelian. The finite [[Simple group|simple]] abelian groups are exactly the cyclic groups of [[Prime number|prime]] [[Order (group theory)|order]].<ref>Rose 2012, [https://books.google.com/books?id=j-I7Zpq3GdIC&pg=PA32 p. 32].</ref>{{rp|32}}
* The concepts of abelian group and <math>\mathbb{Z}</math>-[[Module (mathematics)|module]] agree.  More specifically, every <math>\mathbb{Z}</math>-module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers <math>\mathbb{Z}</math> in a unique way.

In general, [[Matrix (mathematics)|matrices]], even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. However, some groups of matrices are abelian groups under matrix multiplication – one example is the group of <math>2 \times 2</math> [[Rotation matrix|rotation matrices]].

== Historical remarks ==
<!-- This particular statement seems to be suspect, but the direction is right. Note: updated and corrected on Sept. 2, 2012 -->
[[Biography:Camille Jordan|Camille Jordan]] named abelian groups after {{wipe|Norway}} [[Biography:Mathematician|mathematician]] [[Biography:Niels Henrik Abel|Niels Henrik Abel]], as Abel had found that the commutativity of the group of a [[Polynomial|polynomial]] implies that the roots of the polynomial can be calculated by using radicals.<ref>[[Biography:David A. Cox|Cox, D. A.]], ''Galois Theory'' (Hoboken, NJ: [[Company:Wiley (publisher)|John Wiley & Sons]], 2004), [https://books.google.com/books?id=96P8lsAF7fcC&pg=PA144 pp. 144–145].</ref>{{rp|144–145}}<ref>Kepner, J., & H. Jananthan, ''Mathematics of Big Data'' (Cambridge, MA: MIT Press, 2018), [https://google.com/books?id=_rVNEAAAQBAJ&pg=PT157 pp. 157–158].</ref>{{rp|157–158}}

== Properties ==
If <math>n</math> is a [[Natural number|natural number]] and <math>x</math> is an element of an abelian group <math>G</math> written additively, then <math>nx</math> can be defined as <math>x + x + \cdots + x</math> (<math>n</math> summands) and <math>(-n)x = -(nx)</math>. In this way, <math>G</math> becomes a [[Module (mathematics)|module]] over the [[Ring (mathematics)|ring]] <math>\mathbb{Z}</math> of integers. In fact, the modules over <math>\mathbb{Z}</math> can be identified with the abelian groups.<ref>Eklof, Paul C., & Göbel, Rüdiger, eds., '' Abelian Groups and Modules: International Conference in Dublin, August 10–14, 1998'' ([[Place:Basel|Basel]]: [[Company:Springer Science+Business Media|Springer Basel AG]], 1999), [https://books.google.com/books?id=AMcACAAAQBAJ&pg=PA94 pp. 94–97].</ref>{{rp|94–97}}

Theorems about abelian groups (i.e. [[Module (mathematics)|module]]s over the [[Principal ideal domain|principal ideal domain]] <math>\mathbb{Z}</math>) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of [[Finitely generated abelian group|finitely generated abelian group]]s which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a [[Direct sum|direct sum]] of a [[Torsion group|torsion group]] and a [[Free abelian group|free abelian group]]. The former may be written as a direct sum of finitely many groups of the form <math>\mathbb{Z}/p^k\mathbb{Z}</math> for <math>p</math> prime, and the latter is a direct sum of finitely many copies of <math>\mathbb{Z}</math>.

If <math>f, g: G \to H</math> are two [[Group homomorphism|group homomorphism]]s between abelian groups, then their sum <math>f + g</math>, defined by <math>(f + g)(x) = f(x) + g(x)</math>, is again a homomorphism. (This is not true if <math>H</math> is a non-abelian group.) The set <math>\text{Hom}(G,H)</math> of all group homomorphisms from <math>G</math> to <math>H</math> is therefore an abelian group in its own right.

Somewhat akin to the [[Dimension (vector space)|dimension]] of [[Vector space|vector space]]s, every abelian group has a ''[[Rank of an abelian group|rank]]''. It is defined as the maximal [[Cardinal number|cardinality]] of a set of linearly independent (over the integers) elements of the group.<ref>Dixon, M. R., Kurdachenko, L. A., & Subbotin, I. Y., ''Linear Groups: The Accent on Infinite Dimensionality'' (Milton Park, Abingdon-on-Thames & Oxfordshire: Taylor & Francis, 2020), [https://books.google.com/books?id=fODaDwAAQBAJ&pg=PT49 pp. 49–50].</ref>{{rp|49–50}} Finite abelian groups and torsion groups have rank zero, and every abelian group of rank zero is a torsion group. The integers and the [[Rational number|rational number]]s have rank one, as well as every nonzero [[Additive group|additive subgroup]] of the rationals. On the other hand, the [[Multiplicative group|multiplicative group]] of the nonzero rationals has an infinite rank, as it is a free abelian group with the set of the [[Prime number|prime number]]s as a basis (this results from the [[Fundamental theorem of arithmetic|fundamental theorem of arithmetic]]).

The [[Center (group theory)|center]] <math>Z(G)</math> of a group <math>G</math> is the set of elements that commute with every element of <math>G</math>. A group <math>G</math> is abelian if and only if it is equal to its center <math>Z(G)</math>. The center of a group <math>G</math> is always a [[Characteristic subgroup|characteristic]] abelian subgroup of <math>G</math>. If the quotient group <math>G/Z(G)</math> of a group by its center is cyclic then <math>G</math> is abelian.<ref>Rose 2012, [https://books.google.com/books?id=pYk6AAAAIAAJ&pg=PA48 p. 48].</ref>

== Finite abelian groups ==
Cyclic groups of [[Modular arithmetic|integers modulo <math>n</math>]], <math>\mathbb{Z}/n\mathbb{Z}</math>, were among the first examples of groups. It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius and [[Biography:Ludwig Stickelberger|Ludwig Stickelberger]] and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of [[Linear algebra|linear algebra]].

Any group of prime order is isomorphic to a cyclic group and therefore abelian. Any group whose order is a square of a prime number is also abelian.<ref>Rose 2012, [https://books.google.com/books?id=pYk6AAAAIAAJ&pg=PA79 p. 79].</ref> In fact, for every prime number <math>p</math> there are (up to isomorphism) exactly two groups of order <math>p^2</math>, namely <math>\mathbb{Z}_{p^2}</math> and <math>\mathbb{Z}_p\times\mathbb{Z}_p</math>.

=== Classification ===
The '''fundamental theorem of finite abelian groups''' states that every finite abelian group <math>G</math> can be expressed as the direct sum of cyclic subgroups of [[Prime number|prime]]-power order; it is also known as the '''basis theorem for finite abelian groups'''. Moreover, automorphism groups of cyclic groups are examples of abelian groups.<ref>Kurzweil, H., & Stellmacher, B., ''The Theory of Finite Groups: An Introduction'' (New York, Berlin, Heidelberg: [[Company:Springer Science+Business Media|Springer Verlag]], 2004), [https://books.google.com/books?id=iebSBwAAQBAJ&pg=PA43 pp. 43–54].</ref> This is generalized by the [[Fundamental theorem of finitely generated abelian groups|fundamental theorem of finitely generated abelian groups]], with finite groups being the special case when ''G'' has zero [[Rank of an abelian group|rank]]; this in turn admits numerous further generalizations.

The classification was proven by [[Biography:Leopold Kronecker|Leopold Kronecker]] in 1870, though it was not stated in modern group-theoretic terms until later, and was preceded by a similar classification of quadratic forms by [[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]] in 1801; see [[Fundamental theorem of finitely generated abelian groups#History|history]] for details.

The cyclic group <math>\mathbb{Z}_{mn}</math> of order <math>mn</math> is isomorphic to the direct sum of <math>\mathbb{Z}_m</math> and <math>\mathbb{Z}_n</math> if and only if <math>m</math> and <math>n</math> are coprime. It follows that any finite abelian group <math>G</math> is isomorphic to a direct sum of the form

:<math>\bigoplus_{i=1}^{u}\ \mathbb{Z}_{k_i}</math>

in either of the following canonical ways:
* the numbers <math>k_1, k_2, \dots, k_u</math> are powers of (not necessarily distinct) primes,
* or <math>k_1</math> [[Divisor|divides]] <math>k_2</math>, which divides <math>k_3</math>, and so on up to <math>k_u</math>.

For example, <math>\mathbb{Z}_{15}</math> can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: <math>\mathbb{Z}_{15} \cong  \{0,5,10\} \oplus \{0,3,6,9,12\}</math>. The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are [[Group isomorphism|isomorphic]].

For another example, every abelian group of order 8 is isomorphic to either <math>\mathbb{Z}_8</math> (the integers 0 to 7 under addition modulo 8), <math>\mathbb{Z}_4\oplus \mathbb{Z}_2</math> (the odd integers 1 to 15 under multiplication modulo 16), or <math>\mathbb{Z}_2\oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2</math>.

See also [[List of small groups|list of small groups]] for finite abelian groups of order 30 or less.

=== Automorphisms ===
One can apply the fundamental theorem to count (and sometimes determine) the [[Group isomorphism#Automorphisms|automorphisms]] of a given finite abelian group <math>G</math>. To do this, one uses the fact that if <math>G</math> splits as a direct sum <math>H\oplus K</math> of subgroups of coprime order, then
:<math>\operatorname{Aut}(H\oplus K) \cong \operatorname{Aut}(H)\oplus \operatorname{Aut}(K).</math>

Given this, the fundamental theorem shows that to compute the automorphism group of <math>G</math> it suffices to compute the automorphism groups of the [[Sylow theorems|Sylow]] <math>p</math>-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of <math>p</math>). Fix a prime <math>p</math> and suppose the exponents <math>e_i</math> of the cyclic factors of the Sylow <math>p</math>-subgroup are arranged in increasing order:

:<math>e_1\leq e_2 \leq\cdots\leq e_n</math>

for some <math>n > 0</math>. One needs to find the automorphisms of

:<math>\mathbf{Z}_{p^{e_1}} \oplus \cdots \oplus \mathbf{Z}_{p^{e_n}}.</math>

One special case is when <math>n = 1</math>, so that there is only one cyclic prime-power factor in the Sylow <math>p</math>-subgroup <math>P</math>. In this case the theory of automorphisms of a finite [[Cyclic group|cyclic group]] can be used. Another special case is when <math>n</math> is arbitrary but <math>e_i = 1</math> for <math> 1 \le i  \le n</math>. Here, one is considering <math>P</math> to be of the form

:<math>\mathbf{Z}_p \oplus \cdots \oplus \mathbf{Z}_p,</math>

so elements of this subgroup can be viewed as comprising a vector space of dimension <math>n</math> over the finite field of <math>p</math> elements <math>\mathbb{F}_p</math>. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so

:<math>\operatorname{Aut}(P)\cong\mathrm{GL}(n,\mathbf{F}_p),</math>

where <math>\mathrm{GL}</math> is the appropriate [[General linear group|general linear group]]. This is easily shown to have order

:<math> \left|\operatorname{Aut}(P)\right|=(p^n-1)\cdots(p^n-p^{n-1}).</math>

In the most general case, where the <math>e_i</math> and <math>n</math> are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines

:<math>d_k=\max\{r\mid e_r = e_k\}</math>

and

:<math>c_k=\min\{r\mid e_r=e_k\}</math>

then one has in particular <math>k \le d_k</math>, <math>c_k \le k</math>, and

:<math> \left|\operatorname{Aut}(P)\right| = \prod_{k=1}^n (p^{d_k}-p^{k-1}) \prod_{j=1}^n (p^{e_j})^{n-d_j} \prod_{i=1}^n (p^{e_i-1})^{n-c_i+1}. </math>

One can check that this yields the orders in the previous examples as special cases (see Hillar & Rhea).

==Finitely generated abelian groups==
{{main|Finitely generated abelian group}}
An abelian group {{mvar|A}} is finitely generated if it contains a finite set of elements (called ''generators'') <math>G=\{x_1, \ldots, x_n\}</math> such that every element of the group is a [[Linear combination|linear combination]] with integer coefficients of elements of {{mvar|G}}. 

Let {{mvar|L}} be a [[Free abelian group|free abelian group]] with basis <math>B=\{b_1, \ldots, b_n\}.</math> 
There is a unique [[Group homomorphism|group homomorphism]]
<math>p\colon L \to A,</math> such that 
:<math>p(b_i) = x_i\quad \text{for } i=1,\ldots, n.</math>
This homomorphism is [[Surjective function|surjective]], and its [[Kernel (linear algebra)|kernel]] is finitely generated (since integers form a [[Noetherian ring]]). Consider the matrix {{mvar|M}} with integer entries, such that the entries of its {{mvar|j}}th column are the coefficients of the {{mvar|j}}th generator of the kernel. Then, the abelian group is isomorphic to the [[Cokernel|cokernel]] of linear map defined by {{mvar|M}}. Conversely every [[Integer matrix|integer matrix]] defines a finitely generated abelian group.

It follows that the study of finitely generated abelian groups is totally equivalent with the study of integer matrices. In particular, changing the generating set of {{mvar|A}} is equivalent with multiplying {{mvar|M}} on the left by a [[Unimodular matrix|unimodular matrix]] (that is, an invertible integer matrix whose inverse is also an integer matrix). Changing the generating set of the kernel of {{mvar|M}} is equivalent with multiplying {{mvar|M}} on the right by a unimodular matrix.

The [[Smith normal form]] of {{mvar|M}} is a matrix
:<math>S=UMV,</math>
where {{mvar|U}} and {{mvar|V}} are unimodular, and {{mvar|S}} is a matrix such that all non-diagonal entries are zero, the non-zero diagonal entries {{tmath|d_{1,1}, \ldots, d_{k,k} }} are the first ones, and {{tmath|d_{j,j} }} is a divisor of {{tmath|d_{i,i} }} for {{math|''i'' > ''j''}}. The existence and the shape of the Smith normal form proves that the finitely generated abelian group {{mvar|A}} is the [[Direct sum of groups|direct sum]] 
:<math>\Z^r \oplus \Z/d_{1,1}\Z \oplus \cdots \oplus \Z/d_{k,k}\Z,</math> 
where {{mvar|r}} is the number of zero rows at the bottom of {{mvar|S}} (and also the [[Rank of an abelian group|rank]] of the group). This is the [[Fundamental theorem of finitely generated abelian groups|fundamental theorem of finitely generated abelian groups]].

The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums.<ref>Finkelstein, L., & Kantor, W. M., eds., ''Groups and Computation II: Workshop on Groups and Computation, June 7–10, 1995'' (Providence: AMS, 1997), [https://books.google.com/books?id=lyL8Ui36FKsC&pg=PA26&redir_esc=y#v=onepage&q&f=false pp. 26–27].</ref>{{rp|26–27}}

== Infinite abelian groups ==
The simplest infinite abelian group is the infinite cyclic group <math>\mathbb{Z}</math>. Any [[Finitely generated abelian group|finitely generated abelian group]] <math>A</math> is isomorphic to the direct sum of <math>r</math> copies of <math>\mathbb{Z}</math> and a finite abelian group, which in turn is decomposable into a direct sum of finitely many [[Cyclic group|cyclic group]]s of [[Prime power|prime power]] orders. Even though the decomposition is not unique, the number <math>r</math>, called the [[Rank of an abelian group|rank]] of <math>A</math>, and the prime powers giving the orders of finite cyclic summands are uniquely determined.

By contrast, classification of general infinitely generated abelian groups is far from complete. [[Divisible group]]s, i.e. abelian groups <math>A</math> in which the equation <math>nx = a</math> admits a solution <math>x \in A</math> for any natural number <math>n</math> and element <math>a</math> of <math>A</math>, constitute one important class of infinite abelian groups that can be completely characterized. Every divisible group is isomorphic to a direct sum, with summands isomorphic to <math>\mathbb{Q}</math> and [[Prüfer group]]s <math>\mathbb{Q}_p/Z_p</math> for various prime numbers <math>p</math>, and the cardinality of the set of summands of each type is uniquely determined.<ref>For example, <math>\mathbb{Q}/\mathbb{Z} \cong \sum_p \mathbb{Q}_p/\mathbb{Z}_p</math>.</ref> Moreover, if a divisible group <math>A</math> is a subgroup of an abelian group <math>G</math> then <math>A</math> admits a direct complement: a subgroup <math>C</math> of <math>G</math> such that <math>G = A \oplus C</math>. Thus divisible groups are [[Injective module|injective module]]s in the [[Category of abelian groups|category of abelian groups]], and conversely, every injective abelian group is divisible (Baer's criterion). An abelian group without non-zero divisible subgroups is called '''reduced'''.

Two important special classes of infinite abelian groups with diametrically opposite properties are ''torsion groups'' and ''torsion-free groups'', exemplified by the groups <math>\mathbb{Q}/\mathbb{Z}</math> (periodic) and <math>\mathbb{Q}</math> (torsion-free).

=== Torsion groups ===
An abelian group is called '''periodic''' or '''[[Torsion (algebra)|torsion]]''', if every element has finite [[Order (group theory)|order]]. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second [[Prüfer theorems]] state that if <math>A</math> is a periodic group, and it either has a '''bounded exponent''', i.e., <math>nA = 0</math> for some natural number <math>n</math>, or is countable and the [[Height (abelian group)|<math>p</math>-heights]] of the elements of <math>A</math> are finite for each <math>p</math>, then <math>A</math> is isomorphic to a direct sum of finite cyclic groups.<ref>Countability assumption in the second Prüfer theorem cannot be removed: the torsion subgroup of the [[Direct product|direct product]] of the cyclic groups <math>\mathbb{Z}/p^m\mathbb{Z}</math> for all natural <math>m</math> is not a direct sum of cyclic groups.</ref> 
The cardinality of the set of direct summands isomorphic to <math>\mathbb{Z}/p^m\mathbb{Z}</math> in such a decomposition is an invariant of <math>A</math>.<ref>Faith, C. C., ''Rings and Things and a Fine Array of Twentieth Century Associative Algebra'' (Providence: AMS, 2004), [https://books.google.com/books?id=H1TzBwAAQBAJ&pg=PA6 p. 6].</ref>{{rp|6}} These theorems were later subsumed in the '''Kulikov criterion'''. In a different direction, [[Biography:Helmut Ulm|Helmut Ulm]] found an extension of the second Prüfer theorem to countable abelian <math>p</math>-groups with elements of infinite height: those groups are completely classified by means of their Ulm invariants.<ref>Gao, S., ''Invariant Descriptive Set Theory'' (Boca Raton, FL: CRC Press, 2008), [https://google.com/books?id=VfiOZWY282QC&pg=PT317 p. 317].</ref>{{rp|317}}

=== Torsion-free and mixed groups ===
An abelian group is called '''torsion-free''' if every non-zero element has infinite order. Several classes of [[Torsion-free abelian group|torsion-free abelian group]]s have been studied extensively:

* [[Free abelian group]]s, i.e. arbitrary direct sums of <math>\mathbb{Z}</math>
* [[Cotorsion group|Cotorsion]] and [[Algebraically compact module|algebraically compact]] torsion-free groups such as the <math>p</math>-adic integers
* [[Slender group]]s<ref>Albrecht, U., "Products of Slender Abelian Groups", in Göbel, R., & Walker, E., eds., ''Abelian Group Theory: Proceedings of the Third Conference Held on Abelian Group Theory at Oberwolfach, August 11-17, 1985'' (New York: Gordon & Breach, 1987), [https://books.google.com/books?id=kwvGp9_4lOMC&pg=PT259 pp. 259–274].</ref>{{rp|259–274}}

An abelian group that is neither periodic nor torsion-free is called '''mixed'''. If <math>A</math> is an abelian group and <math>T(A)</math> is its [[Torsion subgroup|torsion subgroup]], then the factor group  <math>A/T(A)</math> is torsion-free. However, in general the torsion subgroup is not a direct summand of <math>A</math>, so  <math>A</math> is ''not'' isomorphic to <math>T(A) \oplus A/T(A)</math>. Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group <math>\mathbb{Z}</math> of integers is torsion-free <math>\mathbb{Z}</math>-module.<ref>Lal, R., ''Algebra 2: Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier'' (Berlin, Heidelberg: Springer, 2017), [https://books.google.com/books?id=FwPNDgAAQBAJ&pg=PA206 p. 206].</ref>{{rp|206}}

=== Invariants and classification ===
One of the most basic invariants of an infinite abelian group <math>A</math> is its [[Rank of an abelian group|rank]]: the cardinality of the maximal linearly independent subset of <math>A</math>. Abelian groups of rank 0 are precisely the periodic groups, while [[Torsion-free abelian groups of rank 1|torsion-free abelian groups of rank 1]] are necessarily subgroups of <math>\mathbb{Q}</math> and can be completely described. More generally, a torsion-free abelian group of finite rank <math>r</math> is a subgroup of <math>\mathbb{Q}_r</math>. On the other hand, the group of <math>p</math>-adic integers <math>\mathbb{Z}_p</math> is a torsion-free abelian group of infinite <math>\mathbb{Z}</math>-rank and the groups <math>\mathbb{Z}_p^n</math> with different <math>n</math> are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups.

The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before 1950 and form a foundation of the classification of more general infinite abelian groups. Important technical tools used in classification of infinite abelian groups are [[Pure subgroup|pure]] and [[Basic subgroup|basic]] subgroups. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. See the books by [[Biography:Irving Kaplansky|Irving Kaplansky]], [[Biography:László Fuchs|László Fuchs]], Phillip Griffith, and David Arnold, as well as the proceedings of the conferences on Abelian Group Theory published in ''Lecture Notes in Mathematics'' for more recent findings.

=== Additive groups of rings ===
The additive group of a [[Ring (mathematics)|ring]] is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication). Some important topics in this area of study are:

* [[Tensor product]]
* A.L.S. Corner's results on countable torsion-free groups
* Shelah's work to remove cardinality restrictions
* [[Burnside ring]]

== Relation to other mathematical topics ==
Many large abelian groups possess a natural [[Topology|topology]], which turns them into [[Topological group|topological group]]s.

The collection of all abelian groups, together with the [[Group homomorphism|homomorphisms]] between them, forms the [[Category of abelian groups|category]] <math>\textbf{Ab}</math>, the prototype of an [[Abelian category|abelian category]].

{{harvs|first=Wanda|last=Szmielew|year=1955|txt}} proved that the first-order theory of abelian groups, unlike its non-abelian counterpart, is decidable. Most [[Algebraic structure|algebraic structure]]s other than [[Boolean algebra (structure)|Boolean algebras]] are [[Philosophy:Decidability (logic)|undecidable]].

There are still many areas of current research:
*Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the [[Torsion-free abelian groups of rank 1|rank 1]] case are well understood;
*There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups;
*While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the case of countable mixed groups is much less mature.
*Many mild extensions of the first-order theory of abelian groups are known to be undecidable.
*Finite abelian groups remain a topic of research in [[Computational group theory|computational group theory]].

Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the [[Set theory|set theory]] commonly assumed to underlie all of mathematics. Take the [[Whitehead problem]]: are all Whitehead groups of infinite order also [[Free abelian group|free abelian group]]s? In the 1970s, [[Biography:Saharon Shelah|Saharon Shelah]] proved that the Whitehead problem is:
* Undecidable in ZFC (Zermelo–Fraenkel axioms), the conventional axiomatic set theory from which nearly all of present-day mathematics can be derived. The Whitehead problem is also the first question in ordinary mathematics proved undecidable in ZFC;
* Undecidable even if ZFC is augmented by taking the generalized continuum hypothesis as an axiom;
* Positively answered if ZFC is augmented with the axiom of constructibility (see statements true in L).

== A note on typography ==
Among mathematical adjectives derived from the proper name of a [[Biography:Mathematician|mathematician]], the word "abelian" is rare in that it is often spelled with a lowercase '''a''', rather than an uppercase '''A''', the lack of capitalization being a tacit acknowledgment not only of the degree to which Abel's name has been institutionalized but also of how ubiquitous in modern mathematics are the concepts introduced by him.<ref>{{cite web|url=https://www.maa.org/external_archive/devlin/devlin_04_04.html|archive-url=https://web.archive.org/web/20121231055255/http://www.maa.org/devlin/devlin_04_04.html |archive-date=31 December 2012|url-status=dead|access-date=3 July 2016|title=Abel Prize Awarded: The Mathematicians' Nobel}}</ref>

== See also ==
{{Algebraic structures}}
*{{annotated link|Commutator subgroup}}
*{{annotated link|Dihedral group of order 6}}, the smallest non-abelian group
*{{annotated link|Elementary abelian group}}
*{{annotated link|Grothendieck group}}
*{{annotated link|Pontryagin duality}}

== Notes ==
{{Reflist}}

== References ==
* {{cite book |last=Cox |first=David |url=https://books.google.com/books?id=96P8lsAF7fcC |date=2004 |title=Galois Theory |publisher=Wiley-Interscience |isbn=9781118031339 |mr=2119052 }}
* {{cite book|last=Fuchs|first=László|date=1970 |title=Infinite Abelian Groups |series=Pure and Applied Mathematics |volume=36-I |publisher=[[Company:Academic Press|Academic Press]] |mr=0255673 }}
* {{cite book |last=Fuchs |first=László |date=1973 |title=Infinite Abelian Groups |series=Pure and Applied Mathematics |volume=36-II |publisher=[[Company:Academic Press|Academic Press]] |mr=0349869 }}
* {{cite book |first=Phillip A. |last=Griffith  |date=1970 |title=Infinite Abelian group theory |series=Chicago Lectures in Mathematics |publisher=[[Organization:University of Chicago Press|University of Chicago Press]] |isbn=0-226-30870-7}}
* {{cite book |last=Herstein |first=I. N. |date=1975 |title=Topics in Algebra |url=https://archive.org/details/topicsinalgebra00hers |url-access=registration |edition=2nd |publisher={{wipe|John Wiley & Sons}} |isbn=0-471-02371-X}}
* {{cite journal |last1=Hillar |first1=Christopher |last2=Rhea |first2=Darren |date=2007 |title=Automorphisms of finite abelian groups |url=https://web.archive.org/web/20060629091951id_/http://www.math.tamu.edu:80/~chillar/files/autabeliangrps.pdf|journal=[[American Mathematical Monthly]] |volume=114 |issue=10 |pages=917–923 |doi=10.1080/00029890.2007.11920485 |arxiv=math/0605185|jstor=27642365|bibcode=2006math......5185H |s2cid=1038507 }}
* {{Cite book| last=Jacobson| first=Nathan| date=2009| title=Basic Algebra I |url=https://books.google.com/books?id=JHFpv0tKiBAC| edition=2nd | publisher=[[Social:Dover Publications|Dover Publications]] | isbn = 978-0-486-47189-1}}
* {{cite book |last=Rose |first=John S. |date=2012 |title=A Course on Group Theory |url=https://books.google.com/books?id=pYk6AAAAIAAJ|publisher=Dover Publications |isbn=978-0-486-68194-8}} Unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978.
* {{cite journal |last=Szmielew |first=Wanda|date=1955 |title=Elementary Properties of Abelian Groups |journal=[[Fundamenta Mathematicae]] |volume=41 |issue=2|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm41/fm41122.pdf|pages= 203–271|doi=10.4064/fm-41-2-203-271|mr=0072131|zbl=0248.02049|doi-access=free}}
* {{cite journal |last1=Robinson |first1=Abraham|last2=Zakon |first2=Elias|date=1960 |title=Elementary Properties of Ordered Abelian Groups |journal=[[Organization:Transactions of the American Mathematical Society|Transactions of the American Mathematical Society]] |volume=96 |issue=2|url=https://www.ams.org/journals/tran/1960-096-02/S0002-9947-1960-0114855-0/S0002-9947-1960-0114855-0.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/tran/1960-096-02/S0002-9947-1960-0114855-0/S0002-9947-1960-0114855-0.pdf |archive-date=2022-10-09 |url-status=live|pages= 222–236|doi=10.2307/1993461|jstor=1993461 |doi-access=free}}

==External links==
* {{springer|title=Abelian group|id=p/a010230|mode=cs1}}

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